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Theorem bcthlem2 25292

Description: Lemma for bcth 25296. The balls in the sequence form an inclusion chain. (Contributed by Mario Carneiro, 7-Jan-2014.)

**Hypotheses**
Ref	Expression
bcth.2	⊢ 𝐽 = (MetOpen‘𝐷)
bcthlem.4	⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
bcthlem.5	⊢ 𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ⁺) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))})
bcthlem.6	⊢ (𝜑 → 𝑀:ℕ⟶(Clsd‘𝐽))
bcthlem.7	⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
bcthlem.8	⊢ (𝜑 → 𝐶 ∈ 𝑋)
bcthlem.9	⊢ (𝜑 → 𝑔:ℕ⟶(𝑋 × ℝ⁺))
bcthlem.10	⊢ (𝜑 → (𝑔‘1) = ⟨𝐶, 𝑅⟩)
bcthlem.11	⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)))

**Assertion**
Ref	Expression
bcthlem2	⊢ (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))

Distinct variable groups: 𝑘,𝑛,𝑟,𝑥,𝑧 𝐶,𝑟,𝑥 𝑔,𝑘,𝑛,𝑟,𝑥,𝑧,𝐷 𝑔,𝐹,𝑘,𝑛,𝑟,𝑥,𝑧 𝑔,𝐽,𝑘,𝑛,𝑟,𝑥,𝑧 𝑔,𝑀,𝑘,𝑛,𝑟,𝑥,𝑧 𝜑,𝑘,𝑛,𝑟,𝑥,𝑧 𝑥,𝑅 𝑔,𝑋,𝑘,𝑛,𝑟,𝑥,𝑧 Allowed substitution hints: 𝜑(𝑔) 𝐶(𝑧,𝑔,𝑘,𝑛) 𝑅(𝑧,𝑔,𝑘,𝑛,𝑟)

**Proof of Theorem bcthlem2**
Step	Hyp	Ref	Expression
1		bcthlem.11	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)))
2		fvoveq1 7390	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (𝑔‘(𝑘 + 1)) = (𝑔‘(𝑛 + 1)))
3		id 22	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → 𝑘 = 𝑛)
4		fveq2 6840	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (𝑔‘𝑘) = (𝑔‘𝑛))
5	3, 4	oveq12d 7385	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (𝑘𝐹(𝑔‘𝑘)) = (𝑛𝐹(𝑔‘𝑛)))
6	2, 5	eleq12d 2830	. . . . . 6 ⊢ (𝑘 = 𝑛 → ((𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)) ↔ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛))))
7	6	rspccva 3563	. . . . 5 ⊢ ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)) ∧ 𝑛 ∈ ℕ) → (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)))
8	1, 7	sylan 581	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)))
9		bcthlem.9	. . . . . 6 ⊢ (𝜑 → 𝑔:ℕ⟶(𝑋 × ℝ⁺))
10	9	ffvelcdmda 7036	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑔‘𝑛) ∈ (𝑋 × ℝ⁺))
11		bcth.2	. . . . . . 7 ⊢ 𝐽 = (MetOpen‘𝐷)
12		bcthlem.4	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
13		bcthlem.5	. . . . . . 7 ⊢ 𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ⁺) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))})
14	11, 12, 13	bcthlem1 25291	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (𝑔‘𝑛) ∈ (𝑋 × ℝ⁺))) → ((𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)) ↔ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)))))
15	14	expr 456	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑔‘𝑛) ∈ (𝑋 × ℝ⁺) → ((𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)) ↔ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛))))))
16	10, 15	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)) ↔ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)))))
17	8, 16	mpbid 232	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛))))
18		cmetmet 25253	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
19	12, 18	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
20		metxmet 24299	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
21	19, 20	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
22	11	mopntop 24405	. . . . . . . . . 10 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
23	21, 22	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ Top)
24		xp1st 7974	. . . . . . . . . . . 12 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → (1^st ‘(𝑔‘(𝑛 + 1))) ∈ 𝑋)
25		xp2nd 7975	. . . . . . . . . . . . 13 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → (2^nd ‘(𝑔‘(𝑛 + 1))) ∈ ℝ⁺)
26	25	rpxrd 12987	. . . . . . . . . . . 12 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → (2^nd ‘(𝑔‘(𝑛 + 1))) ∈ ℝ^*)
27	24, 26	jca 511	. . . . . . . . . . 11 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → ((1^st ‘(𝑔‘(𝑛 + 1))) ∈ 𝑋 ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) ∈ ℝ^*))
28		blssm 24383	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (1^st ‘(𝑔‘(𝑛 + 1))) ∈ 𝑋 ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) ∈ ℝ^*) → ((1^st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2^nd ‘(𝑔‘(𝑛 + 1)))) ⊆ 𝑋)
29	28	3expb 1121	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((1^st ‘(𝑔‘(𝑛 + 1))) ∈ 𝑋 ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) ∈ ℝ^*)) → ((1^st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2^nd ‘(𝑔‘(𝑛 + 1)))) ⊆ 𝑋)
30	21, 27, 29	syl2an 597	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → ((1^st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2^nd ‘(𝑔‘(𝑛 + 1)))) ⊆ 𝑋)
31		df-ov 7370	. . . . . . . . . . . 12 ⊢ ((1^st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2^nd ‘(𝑔‘(𝑛 + 1)))) = ((ball‘𝐷)‘⟨(1^st ‘(𝑔‘(𝑛 + 1))), (2^nd ‘(𝑔‘(𝑛 + 1)))⟩)
32		1st2nd2 7981	. . . . . . . . . . . . 13 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → (𝑔‘(𝑛 + 1)) = ⟨(1^st ‘(𝑔‘(𝑛 + 1))), (2^nd ‘(𝑔‘(𝑛 + 1)))⟩)
33	32	fveq2d 6844	. . . . . . . . . . . 12 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) = ((ball‘𝐷)‘⟨(1^st ‘(𝑔‘(𝑛 + 1))), (2^nd ‘(𝑔‘(𝑛 + 1)))⟩))
34	31, 33	eqtr4id 2790	. . . . . . . . . . 11 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → ((1^st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2^nd ‘(𝑔‘(𝑛 + 1)))) = ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))))
35	34	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → ((1^st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2^nd ‘(𝑔‘(𝑛 + 1)))) = ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))))
36	11	mopnuni 24406	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
37	21, 36	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
38	37	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → 𝑋 = ∪ 𝐽)
39	30, 35, 38	3sstr3d 3976	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ∪ 𝐽)
40		eqid 2736	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
41	40	sscls 23021	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ∪ 𝐽) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))))
42	23, 39, 41	syl2an2r 686	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))))
43		difss2 4078	. . . . . . . 8 ⊢ (((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))
44		sstr2 3928	. . . . . . . 8 ⊢ (((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) → (((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))))
45	42, 43, 44	syl2im 40	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → (((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))))
46	45	a1d 25	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → ((2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) → (((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))))
47	46	ex 412	. . . . 5 ⊢ (𝜑 → ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → ((2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) → (((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))))))
48	47	3impd 1350	. . . 4 ⊢ (𝜑 → (((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛))) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))))
49	48	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛))) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))))
50	17, 49	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))
51	50	ralrimiva 3129	1 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3051 ∖ cdif 3886 ⊆ wss 3889 ⟨cop 4573 ∪ cuni 4850 class class class wbr 5085 {copab 5147 × cxp 5629 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ∈ cmpo 7369 1^st c1st 7940 2^nd c2nd 7941 1c1 11039 + caddc 11041 ℝ^*cxr 11178 < clt 11179 / cdiv 11807 ℕcn 12174 ℝ⁺crp 12942 ∞Metcxmet 21337 Metcmet 21338 ballcbl 21339 MetOpencmopn 21342 Topctop 22858 Clsdccld 22981 clsccl 22983 CMetccmet 25221

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-sup 9355 df-inf 9356 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-n0 12438 df-z 12525 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-topgen 17406 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-top 22859 df-topon 22876 df-bases 22911 df-cld 22984 df-cls 22986 df-cmet 25224

This theorem is referenced by: bcthlem3 25293 bcthlem4 25294

W3C validator